Chapter 23 Estimating Volatilities and Correlations. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C.

Transcription

1 Chapter 3 Estatg Volatltes ad Correlatos Copyrght Joh C. Hull 04

2 Stadard Approach to Estatg Volatlty (page 5)! Defe σ as the volatlty per day betwee day - ad day, as estated at ed of day -! Defe S as the value of arket varable at ed of day! Defe u = l(s /S - ) σ = ( u u) = u = = u Copyrght Joh C. Hull 04

3 Splfcatos Usually Made Rsk Maageet (page 5)! Set u = (S S - )/S -! Assue that the ea value of u s zero! Replace by Ths gves σ = u = Copyrght Joh C. Hull 04 3

4 Weghtg Schee Istead of assgg equal weghts to the observatos we ca set σ = = α u = where α = Copyrght Joh C. Hull 04 4

5 ARCH() Model (page 53) I a ARCH() odel we also assg soe weght to the log-ru varace rate, V L : σ γ + = γv where = α L + = α = u Copyrght Joh C. Hull 04 5

6 EWMA Model! I a expoetally weghted ovg average odel, the weghts assged to the u decle expoetally as we ove back through te! Ths leads to σ = λσ + ( λ) u Copyrght Joh C. Hull 04 6

7 To Show that Weghts Decle Expoetally Copyrght Joh C. Hull 04 7 λ λ σ + λ λ + λ + λ + λ λ + λ λ = σ σ σ σ σ + λ λ + λ λ = λ + λ + λσ = λ λ + = λσ σ rate ad decle at at- start Weghts ad so o: the for the for, Substtutg for u u u u u u u u u ) ( ) ( ) ( ) (, ) ( ) ( ) ( ] ) ( [ ) (

8 Attractos of EWMA! Relatvely lttle data eeds to be stored! We eed oly reeber the curret estate of the varace rate ad the ost recet observato o the arket varable! Tracks volatlty chages! 0.94 s a popular choce for λ Copyrght Joh C. Hull 04 8

9 GARCH (,) page 55 I GARCH (,) we assg soe weght to the log-ru average varace rate σ = γv L + αu + βσ Sce weghts ust su to γ + α + β = Copyrght Joh C. Hull 04 9

10 GARCH (,) cotued Settg ω = γv the GARCH (,) odel s σ = ω + αu + βσ ad V L ω = α β Copyrght Joh C. Hull 04 0

11 Exaple (Exaple 3., page 55)! Suppose σ = u σ! The log-ru varace rate s so that the log-ru volatlty per day s.4% Copyrght Joh C. Hull 04

12 Exaple cotued! Suppose that the curret estate of the volatlty s.6% per day ad the ost recet percetage chage the arket varable s %.! The ew varace rate s = The ew volatlty s.53% per day Copyrght Joh C. Hull 04

13 GARCH (p,q) p q = ω + αu + β jσ = j= σ j Copyrght Joh C. Hull 04 3

14 Maxu Lkelhood Methods! I axu lkelhood ethods we choose paraeters that axze the lkelhood of the observatos occurrg Copyrght Joh C. Hull 04 4

15 Exaple! We observe that a certa evet happes oe te te trals. What s our estate of the proporto of the te, p, that t happes?! The probablty of the evet happeg o oe partcular tral ad ot o the others s p( p! We axze ths to obta a axu lkelhood estate. Result: p = 0. (as expected) 9 ) Copyrght Joh C. Hull 04 5

16 Exaple Estate the varace of observatos fro a oral dstrbuto wth ea zero Copyrght Joh C. Hull 04 6 = = = = π u v v u v v u v ) l( exp : Result to axzg : ths s equvalet Takg logarths Maxze :

17 Applcato to GARCH We choose paraeters that axze Copyrght Joh C. Hull 04 7 = = π v u v v u v ) l( exp or

18 S&P 500 Excel Applcato! Start wth tral values of ω, α, ad β! Update varaces! Calculate = l( v u ) v! Use solver to search for values of ω, α, ad β that axze ths objectve fucto! For effcet operato of Solver: set up spreadsheet so that three ubers that are the sae order of agtude are beg searched for Copyrght Joh C. Hull 04 8

19 S&P 500 Excel Applcato (Table 3.) Date Day S u =(S S - )/S - v =σ l(v ) u /v 8-Jul Jul Jul Jul Aug Total 0,8.349 Copyrght Joh C. Hull 04 9

20 The Results (Fgure 3., page 530) Copyrght Joh C. Hull 04 0

21 Varace Targetg! Oe way of pleetg GARCH(,) that creases stablty s by usg varace targetg! We set the log-ru average volatlty equal to the saple varace! Oly two other paraeters the have to be estated Copyrght Joh C. Hull 04

22 How Good s the Model?! The Ljug-Box statstc tests for autocorrelato! We copare the autocorrelato of the u wth the autocorrelato of the u /σ Copyrght Joh C. Hull 04

23 Forecastg Future Volatlty (equato 3.3, page 53) A few les of algebra shows that E[ σ + k ] = V L + ( α + β) k ( σ V L ) The varace rate for a opto exprg o day s k = 0 E [ σ ] + k Copyrght Joh C. Hull 04 3

24 Forecastg Future Volatlty cotued (equato 3.4, page 534) Defe a = l α + β The volatlty per au for a opto lastg T dayss 5 V L + e at at [ V ( 0) V ] L Copyrght Joh C. Hull 04 4

25 S&P Exaple! ω = , α = , β = a = l = Opto Lfe (days) Volatlty (% per au) Copyrght Joh C. Hull 04 5

26 Volatlty Ter Structures! GARCH (,) suggests that, whe calculatg vega, we should shft the log aturty volatltes less tha the short aturty volatltes! Whe stataeous volatlty chages by Δσ(0), volatlty for T-day opto chages by e at at σ( 0) σ( T ) Δσ( 0) Copyrght Joh C. Hull 04 6

27 Results for S&P 500 (Table 3.4)! Whe stataeous volatlty chages by % Opto Lfe (days) Volatlty crease (%) Copyrght Joh C. Hull 04 7

28 Correlatos ad Covaraces (page ) Defe x =(X X - )/X - ad y =(Y Y - )/Y - Also σ x, : daly vol of X calculated o day σ y, : daly vol of Y calculated o day cov : covarace calculated o day The correlato s cov /(σ x, σ y, ) Copyrght Joh C. Hull 04 8

29 Updatg Correlatos! We ca use slar odels to those for volatltes! Uder EWMA cov = λ cov - +(-λ)x - y - Copyrght Joh C. Hull 04 9

30 Postve Fte Defte Codto A varace-covarace atrx, Ω, s terally cosstet f the postve se-defte codto w T Ωw 0 for all vectors w Copyrght Joh C. Hull 04 30

31 Exaple The varace-covarace atrx s ot terally cosstet Copyrght Joh C. Hull 04 3

32 Volatltes ad Correlatos for Four-Idex o Sept 5, 008 wth Equal Weghts DJIA FTSE CAC 40 Nkke 5 DJIA FTSE CAC Nkke Vol. per day (%) DJIA FTSE CAC 40 Nkke Copyrght Joh C. Hull 04 3

33 Volatltes ad Correlatos for Four-Idex o Sept 5, 008 for EWMA ad λ=0.94 DJIA FTSE CAC 40 Nkke 5 DJIA FTSE 0.6 CAC Nkke Vol. per day (%) DJIA FTSE CAC 40 Nkke Copyrght Joh C. Hull 04 33

34 Oe-Day 99% VaR Estates Hstorcal Sulato $53,385 Model Buldg Equal Weghts $7,757 Model Buldg EWMA $47,05 Copyrght Joh C. Hull 04 34